Bryuno Function and the Standard Map

Abstract

For the standard map the homotopically non-trivial invariant curves of rotation number satisfying the Bryuno condition are shown to be analytic in the perturbative parameter, provided the latter is small enough. The radius of convergence of the Lindstedt series -- sometimes called critical function of the standard map -- is studied and the relation with the Bryuno function is derived: the logarithm of the radius of convergence plus twice the Bryuno function is proved to be bounded (from below and from above) uniformily in the rotation number.